The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A142991 a(1) = 1, a(2) = 9, a(n+2) = 9*a(n+1)+(n+1)*(n+3)*a(n). 3
 1, 9, 89, 936, 10560, 127800, 1657080, 22965120, 339252480, 5326819200, 88651670400, 1559600179200, 28929882240000, 564490975104000, 11560712397696000, 247991610230784000, 5561409662613504000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This is the case m = 3 of the general recurrence a(1) = 1, a(2) = 2*m+3, a(n+2) = (2*m+3)*a(n+1) + (n+1)*(n+3)*a(n), which arises when accelerating the convergence of a certain series for the constant log(2). For remarks on the general case see A142988 (m=0). For other cases see A142989 (m=1) and A142990 (m=2). LINKS FORMULA a(n) = (n+2)!*p(n+2)*sum {k = 1..n} (-1)^(k+1)/(k*(k+1)*(k+2)*p(k+1)*p(k+2)), where p(n) = (2*n^3-3*n^2+7*n-3)/15. Recurrence: a(1) = 1, a(2) = 9, a(n+2) = 9*a(n+1)+(n+1)*(n+3)*a(n). The sequence b(n) := 1/2*(n+2)!*p(n+2) satisfies the same recurrence with b(1) = 9, b(2) = 84. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(9+1*3/(9+2*4/(9+3*5/(9+...+(n-1)*(n+1)/9)))), for n >=2. Lim n -> infinity a(n)/b(n) = 1/(9+1*3/(9+2*4/(9+3*5/(9+...+(n-1)*(n+1)/(9+...))))) = 2*sum {k = 1..inf} (-1)^(k+1)/ (k*(k+1)*(k+2)*p(k+1)*p(k+2)) = 167/6 - 40*log(2). MAPLE p := n -> (2*n^3-3*n^2+7*n-3)/15: a := n -> (n+2)!*p(n+2)*sum ((-1)^(k+1)/(k*(k+1)*(k+2)*p(k+1)*p(k+2)), k = 1..n): seq(a(n), n = 1..20); MATHEMATICA RecurrenceTable[{a[1]==1, a[2]==9, a[n+2]==9a[n+1]+(n+1)(n+3)a[n]}, a, {n, 20}] (* Harvey P. Dale, Jul 18 2020 *) CROSSREFS Cf. A142979, A142983, A142988, A142989, A142990. Sequence in context: A138288 A059482 A109002 * A152267 A082147 A095722 Adjacent sequences:  A142988 A142989 A142990 * A142992 A142993 A142994 KEYWORD easy,nonn AUTHOR Peter Bala, Jul 17 2008 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 7 11:32 EDT 2021. Contains 343650 sequences. (Running on oeis4.)